Finding threshold levels when benefit becomes toxicity (positive becomes negative)

[Andrew Kenny]

In a regression analysis I wish to identify the threshold (characterizing an explanatory variable's quantity) above which the effect of a treatment is positive and below which it is negative.

Consider a "peer effects" scenario; addition of happy prosocial people make parties more fun, while addition of adequately noxious "bad boys" truly can wreck them for others. Restated, imagine that greater average pro-sociality of students at a dorm party has a positive effect on individual ratings of party fun-ness. Again, hypothesize that there is a *threshold* above which the presence of a person makes the party more fun and below which their presence makes it less fun.

How does one identify that threshold?

The principal challenge is how to operationalize the independent variable.

I don't believe a combination of dummies would work, since the coefficient signs are defined by the omitted group, rather than indicating
whether the amount of the variable makes the party objectively better or worse. What I want from a naïve standpoint is dummy variables with no omitted group, but that is not conceivable.

Note: I believe that something like my question is akin to the study of "threshold of toxicity" in medicine, if that helps conceptualize my notion of objective harms and objective benefits, but the study in fact involves characteristics of people in social dynamics.

[Clyde Schechter]

When studying things like threshold of toxicity of a drug, you would not use dummy variables of any kind to represent the dosage. You would use a continuous variable that reflects the actual dose. You would then model the outcome as a function of dose (in the simplest, perhaps overly simple, case a linear function), and then you have an equation relating the two. You can differentiate that to get the marginal effect of a dose increase, and then solve that equation to find the value(s) of dosage for which marginal effect equals 0.

This is all in general terms. To be more specific, you would have to posit a specific functional model relating "number of people at the party" (as a continuous variable) and party fun.

[Andrew Kenny]

This is very interesting, but new territory. The bottle-neck for me is probably the mathematical logic. What could I read to drill this deeper into my synapses?

As one might imagine, my prior ways of thinking were very coefficient centric.

Does differentiate in this sense imply calculus mathematics by hand?

This approximates the model:

Individual-level fun = constant + (Beta)(individual-pro-sociality) + (Beta)mean(group level)pro-sociality

where the data contain many groups

The reason for the group-mean variable is that an anti-social person probably is a negative effect on others in a way separate from effects on himself.

[Clyde Schechter]

Well, if you have a linear model like this, then there is no point where the marginal effect becomes zero. There may be values where the individual-level fun (however you measure that) itself becomes 0--and perhaps that is what you are after. But the marginal effect in a linear model is always constant.

If you are interested in points where individual level fun becomes zero, and you like your linear model, then you have a linear equation:

beta_individual*individual pro-sociality + beta_group * group pro-sociality + constant = 0

Presumably prior to this point you have fit the original model
Individual-level fun = constant + (Beta)(individual-pro-sociality) + (Beta)mean(group level)pro-sociality and arrived at the values of the betas and the constant. So in our new equation, the betas are constants nd the values of individual pro_sociality and group pro_sociality are unknowns we want to solve for.

Now, when you have two variables and only a single linear equation, there are infinitely many solutions, and they all lie on a line in the individual pro_sociality X pro-sociality plane. The equation of that line is given by:

individual pro_sociality = -constant - group pro_sociality * (beta_group/beta_individual).

Whenever the valuesof individual and group prosociality satisfy this relationship (with the values of constant and the two betas that arose from fitting your original model), the expected value of individual-level fun is zero.

[Andrew Kenny]

I see - thank you. If my model doesn't allow zero marginal effects, what model would? Is it a matter of including a fancy transformation of one of the independent variables?

I am ultimately drawn the idea of opposite "results" for different levels. If Paul McCartney comes to the party, the party will get funner, while if the Terminator comes, it will get worse. If what I have in mind is analogous to medicine toxicology, then I suspect the question is analytically coherent but part of me wonders if the idea requires consideration of the distance between the depvar and the independent var.

[Clyde Schechter]

In mathematical terms, if the functional relationship were non-monotonic and have a continuous first derivative that would assure the existence of a critical point (i.e. a point at which there is a zero marginal effect). This condition is sufficient, but not necessary. But the family of functions that satisfy this condition is quite large and includes quadratics, trigonometric functions, and many others. And you would be well advised to seek a model from this family because, a) if the function were monotonic your point of zero marginal effect would represent just a point of inflection, not a turning point, and b) functions that are not differentiable with continuous derivative are difficult to work with. (Of course, if you knew the true model didn't satisfy this condition you might breech it, but even then, all models compromise realism for tractability/simplicity. So one might prefer an approximately correct model that satisfied the condition to an unworkable but "correct" model.)